Question 96
Let $O$ be the circumcenter of triangle $ABC$. Points $A'$, $B'$, $C'$ lie on the segments $BC$, $CA$, $AB$ respectively such that the circumcircles of triangles $AB'C'$, $BC'A'$, and $CA'B'$ pass through $O$. Denote by $l_a$ the radical axis of the circle with center $B'$ and radius $B'C$ and the circle with center $C'$ and radius $C'B$. Define $l_b$ and $l_c$ similarly. Prove that the lines $l_a$, $l_b$, $l_c$ form a triangle such that its orthocenter coincides with the orthocenter of triangle $ABC$.